# [R] How to solve A'A=S for A

Peter Dalgaard BSA p.dalgaard at biostat.ku.dk
Tue Feb 18 23:52:10 CET 2003

```Spencer Graves <spencer.graves at pdf.com> writes:

> I think it's interesting that solve(chol(W)) is not the same as
> chol(solve(W)), but I'll let someone else comment on that.

That's basically because inv(R'R) = inv(R)inv(R') = inv(R)inv(R)'
Notice that the last form is upper tri by lower tri, whereas the
Choleski factorization is the other way around. Actually, this UU'
factorization works just like the Choleski factorization taking the
rows and columns of W in the reverse order so we have

> solve(chol(W))
[,1] [,2] [,3]
[1,]    1  0.0 -2.0
[2,]    0  0.5 -0.5
[3,]    0  0.0  1.0

> zapsmall(chol(solve(W)[3:1,3:1]))[3:1,3:1]
[,1] [,2] [,3]
[1,]    1  0.0    0
[2,]    0  0.5    0
[3,]   -2 -0.5    1

which are each others transposes.

--
O__  ---- Peter Dalgaard             Blegdamsvej 3
c/ /'_ --- Dept. of Biostatistics     2200 Cph. N
(*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907

```